AGT関係式(3) 一般化に向けて

(String Advanced Lectures No.20)

高エネルギー加速器研究機構(KEK)

素粒子原子核研究所(IPNS)

柴 正太郎

2010年6月23日（水） 12:30-14:30

Contents

1. AGT relation for SU(2) quiver theory

2. Partition function of SU(N) quiver theory

3. Toda theory and W-algebra

4. Generalized AGT relation for SU(N) case

5. Towards AdS/CFT duality of AGT relation

AGT relation for SU(2) quiver

We now consider only the linear quiver gauge theories in AGT relation.

Gaiotto’s discussion

An example : SW curve is a sphere with multiple punctures.

The Seiberg-Witten curve in this case corresponds to

4-dim N=2 linear quiver SU(2) gauge theory.

Nekrasov instanton partition function

where equals to the conformal block of

Virasoro algebra with for the vertex operators which are inserted

at z=

Liouville correlation function (corresponding n+3-point function)

where is Nekrasov’s full partition function.

(↑ including 1-loop part)

U(1) part

[Alday-Gaiotto-Tachikawa ’09]AGT relation : SU(2) gauge theory Liouville theory !

Gauge theory Liouville theory

coupling const. position of punctures

VEV of gauge fields internal momenta

mass of matter fields external momenta

1-loop part DOZZ factors

instanton part conformal blocks

deformation parameters Liouville parameters

4-dim theory : SU(2) quiver gauge theory

2-dim theory : Liouville (A1 Toda) field theory

In this case, the 4-dim theory’s partition function Z and the 2-dim theory’s

correlation function correspond to each other :

central charge :

Now we calculate Nekrasov’s partition function of 4-dim SU(N) quiver gauge

theory as the quantity of interest.

SU(2) case : We consider only SU(2)×…×SU(2) quiver gauge theories.

SU(N) case : According to Gaiotto’s discussion, we consider, in general, the

cases of SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2) x SU(d’1) group,

where is non-negative.

SU(N) partition function

Nekrasov’s partition function of 4-dim gauge theory

xx xx

x

*

…x

*

…

…

d’3–d’2d’2–d’1d’1… …

……

…

d3–d2

d2–d1

d1… ………

1-loop part of partition function of 4-dim quiver gauge theory

We can obtain it of the analytic form :

where each factor is defined as

: each factor is a product of double Gamma function!

,

gauge antifund. bifund. fund.

mass massmass

flavor symm. of bifund. is U(1)

VEV

# of d.o.f. depends on dk

deformation parameters

We obtain it of the expansion form of instanton number :

where : coupling const. and

and

Instanton part of partition function of 4-dim quiver gauge theory

Young tableau

< Young tableau >

instanton # = # of boxes

leg

arm

Naive assumption is 2-dim AN-1 Toda theory, since Liouville theory is nothing but

A1 Toda theory. This means that the generalized AGT relation seems

Difference from SU(2) case…

• VEV’s in 4-dim theory and momenta in 2-dim theory have more than one d.o.f.

In fact, the latter corresponds to the fact that the punctures are classified with

more than one kinds of N-box Young tableaux :

< full-type > < simple-type > < other types >

(cf. In SU(2) case, all these Young tableaux become ones of the same type .)

• In general, we don’t know how to calculate the conformal blocks of Toda theory.

……

…

…

………

What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory?

Action :

Toda field with :

It parametrizes the Cartan subspace of AN-1 algebra.

simple root of AN-1 algebra :

Weyl vector of AN-1 algebra :

metric and Ricci scalar of 2-dim surface

interaction parameters : b (real) and

central charge :

Toda theory and W-algebra

What is AN-1 Toda theory? : some extension of Liouville theory

• In this theory, there are energy-momentum tensor and higher spin fields

as Noether currents.

• The symmetry algebra of this theory is called W-algebra.

• For the simplest example, in the case of N=3, the generators are defined as

And, their commutation relation is as follows:

which can be regarded as the extension of Virasoro algebra, and where

,

What is AN-1 Toda field theory? (continued)

We ignore Toda potential

(interaction) at this stage.

• The primary fields are defined as ( is called ‘momentum’) .

• The descendant fields are composed by acting / on the primary

fields as uppering / lowering operators.

• First, we define the highest weight state as usual :

Then we act lowering operators on this state, and obtain various descendant

fields as .

• However, some linear combinations of descendant fields accidentally satisfy

the highest weight condition. They are called null states. For example, the null

states in level-1 descendants are

• As we will see next, we found the fact that these null states in W-algebra are

closely related to the singular behavior of Seiberg-Witten curve near the

punctures. That is, Toda fields whose existence is predicted by AGT relation

may in fact describe the form (or behavior) of Seiberg-Witten curve.

As usual, we compose the primary, descendant, and null fields.

• As we saw, Seiberg-Witten curve is generally represented as

and Laurent expansion near z=z0 of the coefficient function is generally

• This form is similar to Laurent expansion of W-current (i.e. W-generators)

• Also, the coefficients satisfy similar equations, except full-type puncture’s case

This correspondence becomes exact, in some kind of ‘classical’ limit:

(which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?)

• This fact strongly suggests that vertex operators corresponding non-full-type

punctures must be the primary fields which has null states in their descendants.

The singular behavior of SW curve is related to the null fields of W-algebra.[Kanno-Matsuo-SS-Tachikawa ’09]

null condition

~ direction of D4 ~ direction of NS5

• If we believe this suggestion, we can conjecture the form of

momentum of Toda field in vertex operators ,

which corresponds to each kind of punctures.

• To find the form of vertex operators which have the level-1 null state, it is

useful to consider the screening operator (a special type of vertex operator)

• We can show that the state satisfies the highest weight

condition, since the screening operator commutes with all the W-generators.

(Note a strange form of a ket, since the screening operator itself has non-zero momentum.)

• This state doesn’t vanish, if the momentum satisfies

for some j. In this case, the vertex operator has a null state at level .

The punctures on SW curve corresponds to the ‘degenerate’ fields![Kanno-Matsuo-SS-Tachikawa ’09]

• Therefore, the condition of level-1 null state becomes for some j.

• It means that the general form of mometum of Toda fields satisfying this null

state condition is .

Note that this form naturally corresponds to Young tableaux .

• More generally, the null state condition can be written as

(The factors are abbreviated, since they are only the images under Weyl transformation.)

• Moreover, from physical state condition (i.e. energy-momentum is real), we

need to choose , instead of naive generalization:

Here, is the same form of β,

is Weyl vector,

and .

The punctures on SW curve corresponds to the ‘degenerate’ fields!

Generalized AGT relation

Natural form : former’s partition function and latter’s correlation function

Problems and solutions for its proof

• correspondence between each kind of punctures and vertices:

we can conjecture it, using level-1 null state condition.

< full-type > < simple-type > < other types >

• difficulty for calculation of conformal blocks: null state condition resolves it again!

[Wyllard ’09]

[Kanno-Matsuo-SS-Tachikawa ’09]

……

…

…

………

Correspondence : 4-dim SU(N) quiver gauge and 2-dim AN-1 Toda theory

• We put the (primary) vertex operators at punctures, and consider

the correlation functions of them:

• In general, the following expansion is valid:

where

and for level-1 descendants,

: Shapovalov matrix

• It means that all correlation functions consist of 3-point functions and inverse

Shapovalov matrices (propagator), where the intermediate states (descendants)

can be classified by Young tableaux.

On calculation of correlation functions of 2-dim AN-1 Toda theory

descendants

primaries

In fact, we can obtain it of the factorization form of 3-point functions and inverse

Shapovalov matrices :

3-point function : We can obtain it, if one entry has a null state in level-1!

where

highest weight

~ simple punc.

On calculation of correlation functions of 2-dim AN-1 Toda theory

’

Case of SU(3) quiver gauge theory

SU(3) : already checked successfully. [Wyllard ’09] [Mironov-Morozov ’09]

SU(3) x … x SU(3) : We have checked successfully. [Kanno-Matsuo-SS ’10]

SU(3) x SU(2) : We could check it, but only for restricted cases. [Kanno-Matsuo-SS ’10]

Case of SU(4) quiver gauge theory

• In this case, there are punctures which are not full-type nor simple-type.

• So we must discuss in order to check our conjucture (of the simplest example).

• The calculation is complicated because of W4 algebra, but is mostly streightforward.

Case of SU(∞) quiver gauge theory

• In this case, we consider the system of infinitely many M5-branes, which may relate to

AdS dual system of 11-dim supergravity.

• AdS dual system is already discussed using LLM’s droplet ansatz, which is also governed

by Toda equation. [Gaiotto-Maldacena ’09] → subject of next talk…

Our plans of current and future research on generalized AGT relation

Towards AdS/CFT of AGT

CFT side : 4-dim SU(N) quiver gauge theory and 2-dim AN-1Toda theory

• 4-dim theory is conformal.

• The system preserves eight supersymmetries.

AdS side : the system with AdS5 and S2 factor and eight supersymmetries

• This is nothing but the analytic continuation of LLM’s system in M-theory.

• Moreover, when we concentrate on the neighborhood of punctures on

Seiberg-Witten curve, the system gets the

additional S1 ~ U(1) symmetry.

• According to LLM’s discussion, such system can

be analyzed using 3-dim electricity system:

[Lin-Lunin-Maldacena ’04]

[Gaiotto-Maldacena ’09]